Download Quantum Phase Transitions - Subir Sachdev

Quantum Phase Transitions
Subir Sachdev
Department of Physics
Harvard University
Cambridge, MA 02138
USA
E-mail: subir [email protected]
March 21, 2006
To appear in
Handbook of Magnetism and Advanced Magnetic Materials,
edited by H. Kronmuller and S. Parkin.
1
Introduction
The study of quantum phase transitions has been a major focus of theoretical
and experimental work in systems of correlated electrons and in correlated ultracold atoms in recent years. However, some of the best characterized and
understood examples of quantum phase transitions are found in magnetic
materials. These therefore serve as a valuable laboratory for testing our understanding of real systems in the vicinity of quantum critical points. We will
review some of the simplest model systems below, along with their experimental realizations. This will be followed by a discussion of recent theoretical
advances on some novel quantum critical points displayed by quantum magnets which have no direct analog in the theory of classical phase transitions
at finite temperature (T ). Portions of this article have been adapted from
another recent review by the author [a].
1
Quantum Phase Transitions
2
In all the magnetic systems considered below, there is at least one ground
state in which the symmetry of spin rotations is broken. So in this phase we
have
hŜjα i 6= 0.
(1)
at T = 0. Here Ŝ is the electron spin operator on site j, and α = x, y, z
are the spin components. In all cases we consider here the Hamiltonian has
at least a symmetry of spin inversion, and this symmetry is broken by (1).
We are quite familiar with such magnetic systems, as all antiferromagnets,
ferromagnets, or even spin glasses obey (1) at sufficiently low temperatures.
Let us now try to access a paramagnetic phase where
hŜjα i = 0.
(2)
Normally, we do this by raising temperature. The resulting phase transition
between phases characterized by (1) and (2) is well understood, and described
by the well-developed theory of classical phase transitions. This shall not be
our interest here. Rather, we are interested in moving from magnetic system
obeying (1), to a quantum paramagnet obeying (2), by varying a system
parameter at T = 0. There are many experimental and theoretical examples
of such transitions: at the critical point, there is a qualitative change in the
nature of the quantum wavefunction of the ground state.
One crucial feature of quantum phase transitions is that (2) is usually not
sufficient to characterize the paramagnetic phase. In the Landau-GinzburgWilson (LGW) approach to classical phase transitions, one focuses on the
broken symmetry associated with (1), and defines a corresponding order parameter. A field theory of thermal fluctuations of this order parameter is
then sufficient to describe the transition to the paramagnetic phase, and also
to completely characterize the paramagnet. As we will discuss in Section 4,
this procedure is not sufficient in some of the most interesting and physically
important quantum phase transitions. The paramagnetic phase is not completely characterized by (2), and typically breaks some other symmetry of
the Hamiltonian, or has a more subtle ‘topological’ order. Furthermore, this
additional ‘order’ of the paramagnet plays an important role in the theory
of the quantum critical point.
We will begin in Section 2 by introducing some simple lattice models, and
their experimental realizations, which exhibit quantum phase transitions.
The theory of the critical point in these models is based upon a natural
Quantum Phase Transitions
3
extension of the LGW method, and this will be presented in Section 3. This
section will also describe the consequences of a zero temperature critical
point on the non-zero temperature properties. Section 4 will consider more
complex models in which quantum interference effects play a more subtle
role, and which cannot be described in the LGW framework: such quantum
critical points are likely to play a central role in understanding many of the
correlated electron systems of current interest.
2
2.1
Simple models
Ising ferromagnet in a transverse field
This quantum phase transition is realized [b] in the insulator LiHoF4 . The
Ho ion has a S = 1/2 Ising spin which prefers to orient itself either parallel or
anti-parallel to a particular crystalline axis (say z). These Ising spins interact
via the magnetic dipolar coupling and normally form a ferromagnetic ground
state which obeys (1) for α = z. As we describe below, upon application of a
(transverse) magnetic field in the plane perpendicular to the z axis, quantum
fluctuations of the Ising spin are enhanced, and there is eventually a quantum
phase transition to a paramagnetic state obeying (2) for α = z.
Rather than explore the full complexity of the experimentally relevant model,
we will restrict our attention to a simple one-dimensional model, with nearest
neighbor couplings, which displays much of the same physics. The dynamics
of this quantum Ising spin chain is described by the simple Hamiltonian
HI = −J
N
−1
X
j=1
z
σ̂jz σ̂j+1
− gJ
N
X
σ̂jx ,
(3)
j=1
where σ̂jα are the Pauli matrices which act on the Ising spin degrees of freedom
(Ŝjα ∝ σ̂jα ) J > 0 is the ferromagnetic coupling between nearest neighbor
spins, and g ≥ 0 is a dimensionless coupling constant which determines the
strength of the transverse field. In the thermodynamic limit (N → ∞), the
ground state of HI exhibits a second-order quantum phase transition as g is
tuned across a critical value g = gc (for the specific case of HI it is known
that gc = 1), as we will now illustrate.
Quantum Phase Transitions
4
First, consider the ground state of HI for g ¿ 1. At g = 0, there are two
degenerate ferromagnetically ordered ground states
|⇑i =
N
Y
|↑ij ; |⇓i =
j=1
N
Y
|↓ij
(4)
j=1
Each of these states breaks a discrete ‘Ising’ symmetry of the Hamiltonian–
rotations of all spins by 180 degrees about the x axis. These states are more
succinctly characterized by defining the ferromagnetic moment, N0 by
N0 = h⇑| σ̂jz |⇑i = − h⇓| σ̂jz |⇓i
(5)
At g = 0 we clearly have N0 = 1. A key point is that in the thermodynamic
limit, this simple picture of the ground state survives for a finite range of small
g (indeed, for all g < gc ), but with 0 < N0 < 1. The quantum tunnelling
between the two ferromagnetic ground states is exponentially small in N (and
so can be neglected in the thermodynamic limit), and so the ground state
remains two-fold degenerate and the discrete Ising symmetry remains broken.
The change in the wavefunctions of these states from Eq. (4) can be easily
determined by perturbation theory in g: these small g quantum fluctuations
reduce the value of N0 from unity but do not cause the ferromagnetism to
disappear.
Now consider the ground state of HI for g À 1. At g = ∞ there is a single
non-degenerate ground state which fully preserves all symmetries of HI :
|⇒i = 2
−N/2
N ³
Y
´
|↑ij + |↓ij .
(6)
j=1
It is easy to verify that this state has no ferromagnetic moment N0 =
h⇒| σ̂jz |⇒i = 0. Further, perturbation theory in 1/g shows that these features of the ground state are preserved for a finite range of large g values
(indeed, for all g > gc ). One can visualize this ground state as one in which
strong quantum fluctuations have destroyed the ferromagnetism, with the
local magnetic moments quantum tunnelling between ‘up’ and ‘down’ on a
time scale of order ~/J.
Given the very distinct signatures of the small g and large g ground states,
it is clear that the ground state cannot evolve smoothly as a function of g.
Quantum Phase Transitions
5
These must be at least one point of non-analyticity as a function of g: for
HI it is known that there is only a single non-analytic point, and this is at
the location of a second-order quantum phase transition at g = gc = 1.
The character of the excitations above the ground state also undergoes a
qualitative change across the quantum critical point. In both the g < gc and
g > gc phase these excitations can be described in the Landau quasiparticle
scheme i.e. as superpositions of nearly independent particle-like excitations;
a single well-isolated quasiparticle has an infinite lifetime at low excitation
energies. However, the physical nature of the quasiparticles is very different
in the two phases. In the ferromagnetic phase, with g < gc , the quasiparticles
are domain walls between regions of opposite magnetization:
|j, j + 1i =
j
Y
k=1
|↑ik
N
Y
|↓i`
(7)
`=j+1
This is the exact wavefunction of a stationary quasiparticle excitation between sites j and j + 1 at g = 0; for small non-zero g the quasiparticle
acquires a ‘cloud’ of further spin-flips and also becomes mobile. However the
its qualitative interpretation as a domain wall between the two degenerate
ground states remains valid for all g < gc . In contrast, for g > gc , there is no
ferromagnetism, and the non-degenerate paramagnetic state has a distinct
quasiparticle excitation:
³
´Y
|ji = 2−N/2 |↑ij − |↓ij
(|↑ik + |↓ik ) .
(8)
k6=j
This is a stationary ‘flipped spin’ quasiparticle at site j, with its wavefunction
exact at g = ∞. Again, this quasiparticle is mobile and applicable for all
g > gc , but there is no smooth connection between Eq. (8) and (7).
2.2
Coupled dimer antiferromagnet
Now we consider a model of S = 1/2 spins which interact via an antiferromagnetic exchange, and the Hamiltonian has full SU(2) spin rotation invariance. Physically, the cuprates are by far the most important realization of
Hamiltonians in this class. However, rather than facing the daunting complexity of those compounds, it is useful to study simpler insulators in which
Quantum Phase Transitions
6
Figure 1: The coupled dimer antiferromagnet. Qubits (i.e. S = 1/2 spins) are
placed on the sites, the A links are shown as full lines, and the B links as dashed
lines.
a quantum phase transition from an antiferromagnet to a paramagnet can be
explored. One experimentally and theoretically well studied system [c, d, e, f]
is TlCuCl3 . Here the S = 1/2 spins reside on the Cu+ ions, which reside in
a rather complicated spatial arrangement. As in Section 2.1, we will not explore the full complexity of the experimental magnet, but be satisfied with a
caricature that captures the essential physics. The most important feature of
the crystal structure of TlCuCl3 (as will become clear in Section 4) is that it
is naturally dimerized i.e. there is a pairing between Cu spins which respects
all symmetries of the crystal structure. So will consider the simplest dimer
antiferromagnet of S = 1/2 spins which exhibits a quantum phase transition
essentially equivalent to that found in TlCuCl3 .
The Hamiltonian of the dimer antiferromagnet is illustrated in Fig 1 and is
given by
X ¡
¢
Hd = J
σ̂jx σ̂kx + σ̂jy σ̂ky + σ̂jz σ̂kz
hjki∈A
+
¢
J X ¡ x x
σ̂j σ̂k + σ̂jy σ̂ky + σ̂jz σ̂kz ,
g
(9)
hjki∈B
where now J > 0 is the antiferromagnetic exchange constant, g ≥ 1 is the
dimensionless coupling, and the set of nearest-neighbor links A and B are
defined in Fig 1. An important property of Hd is that it is now invariant
under the full SU(2) group of spin rotations under which the σ̂ α transform as
ordinary vectors (in contrast to the Z2 symmetry group of HI ). In analogy
with HI , we will find that Hd undergoes a quantum phase transition from
a paramagnetic phase which preserves all symmetries of the Hamiltonian at
Quantum Phase Transitions
=(
7
-
)/ /2
Figure 2: The paramagnetic state of Hd for g > gc . The state illustrated is the
exact ground state for g = ∞, and it is adiabatically connected to the ground state
for all g > gc .
large g, to an antiferromagnetic phase which breaks the SU(2) symmetry at
small g. This transition occurs at a critical value g = gc , and the best current
numerical estimate is [g] 1/gc = 0.52337(3).
As in the previous subsection, we can establish the existence of such a quantum phase transition by contrasting the disparate physical properties at large
g with those at g ≈ 1. At g = ∞ the exact ground state of Hd is
´
Y 1 ³
√ |↑ij |↓ik − |↓ij |↑ik
(10)
|spin gapi =
2
hjki∈A
and is illustrated in Fig 2. This state is non-degenerate and invariant under
spin rotations, and so is a paramagnet: the qubits are paired into spin singlet
valence bonds across all the A links. The excitations above the ground state
are created by breaking a valence bond, so that the pair of spins form a spin
triplet with total spin S = 1 — this is illustrated in Fig 3. It costs a large
energy to create this excitation, and at finite g the triplet can hop from link
to link, creating a gapped triplon quasiparticle excitation. This is similar
to the large g paramagnet for HI , with the important difference that each
quasiparticle is now 3-fold degenerate.
At g = 1, the ground state of Hd is not known exactly. However, at this
point Hd becomes equivalent to the nearest-neighbor square lattice antiferromagnet, and this is known to have antiferromagnetic order in the ground
state, as illustrated in Fig 4. This state is similar to the ferromagnetic ground
Quantum Phase Transitions
8
Figure 3: The triplon excitation of the g > gc paramagnet. The stationary triplon
is an eigenstate only for g = ∞ but it becomes mobile for finite g.
Figure 4: Schematic of the ground state with antiferromagnetic order with g < gc .
Quantum Phase Transitions
9
state of HI , with the difference that the magnetic moment now acquires a
staggered pattern on the two sublattices, rather than the uniform moment
of the ferromagnet. Thus in this ground state
hAF| σ̂jα |AFi = N0 ηj nα
(11)
where 0 < N0 < 1 is the antiferromagnetic (or Néel) moment, ηj = ±1 identifies the two sublattices in Fig 4, and nα is an arbitrary unit vector specifying
the orientation of the spontaneous magnetic moment which breaks the O(3)
spin rotation invariance of Hd . The excitations above this antiferromagnet
are also distinct from those of the paramagnet: they are a doublet of spin
waves consisting of a spatial variation in the local orientation, nα , of the
antiferromagnetic order: the energy of this excitation vanishes in the limit of
long wavelengths, in contrast to the finite energy gap of the triplon excitation
of the paramagnet.
As with HI , we can conclude from the distinct characters of the ground states
and excitations for g À 1 and g ≈ 1 that there must be a quantum critical
point at some intermediate g = gc .
3
Quantum criticality
The simple considerations of Section 2 have given a rather complete description (based on the quasiparticle picture) of the physics for g ¿ gc and g À gc .
We turn, finally, to the region g ≈ gc . For the specific models discussed in
Section 2, a useful description is obtained by a method that is a generalization of the LGW method developed earlier for thermal phase transitions.
However, some aspects of the critical behavior (e.g. the general forms of
Eqns (14), (15), and (16)) will apply also to the quantum critical point of
Section 4.
Following the canonical LGW strategy, we need to identify a collective order
parameter which distinguishes the two phases. This is clearly given by the
ferromagnetic moment in Eq. (5) for the quantum Ising chain, and the antiferromagnetic moment in Eq. (11) for the coupled dimer antiferromagnet.
We coarse-grain these moments over some finite averaging region, and at
long wavelengths this yields a real order parameter field φα , with the index
α = 1 . . . n. For the Ising case we have n = 1 and φα is a measure of the local
Quantum Phase Transitions
10
average of N0 as defined in Eq. (5). For the antiferromagnet, a extends over
the three values x, y, z (so n = 3), and three components of φα specify the
magnitude and orientation of the local antiferromagnetic order in Eq. (11);
note the average orientation of a specific spin at site j is ηj times the local
value of φα .
The second step in the LGW approach is to write down a general field theory
for the order parameter, consistent with all symmetries of the underlying
model. As we are dealing with a quantum transition, the field theory has to
extend over spacetime, with the temporal fluctuations representing the sum
over histories in the Feynman path integral approach. With this reasoning,
the proposed partition function for the vicinity of the critical point takes the
following form
· Z
µ
Z
¢
1¡
d
Zφ =
Dφα (x, τ ) exp − d xdτ
(∂τ φα )2 + c2 (∇x φα )2 + sφ2α
2
¸
u ¡ 2 ¢2 ´
+
φ
.
(12)
4! α
Here τ is imaginary time, there is an implied summation over the n values
of the index a, c is a velocity, and s and u > 0 are coupling constants. This
is a field theory in d + 1 spacetime dimensions, in which the Ising chain
corresponds to d = 1 and the dimer antiferromagnet to d = 2. The quantum
phase transition is accessed by tuning the “mass” s: there is a quantum
critical point at s = sc and the s < sc (s > sc ) regions corresponds to the
g < gc (g > gc ) regions of the lattice models. The s < sc phase has hφα i 6= 0
and this corresponds to the spontaneous breaking of spin rotation symmetry
noted in Eqs. (5) and (11) for the lattice models. The s > sc phase is the
paramagnet with hφα i = 0. The excitations in this phase can be understood
as small harmonic oscillations of φα about the point (in field space) φα = 0. A
glance at Eqn (12) shows that there are n such oscillators for each wavevector.
These oscillators clearly constitute the g > gc quasiparticles found earlier in
Eqn (8) for the Ising chain (with n = 1) and the triplon quasiparticle (with
n = 3) illustrated in Fig 3) for the dimer antiferromagnet.
We have now seen that there is a perfect correspondence between the phases
of the quantum field theory Zφ and those of the lattice models HI and Hd .
The power of the representation in Eqn. (12) is that it also allows us to get a
simple description of the quantum critical point. In particular, readers may
already have noticed that if we interpret the temporal direction τ in Eqn. (12)
Quantum Phase Transitions
11
as another spatial direction, then Zφ is simply the classical partition function
for a thermal phase transition in a ferromagnet in d + 1 dimensions: this is
the canonical model for which the LGW theory was originally developed. We
can now take over standard results for this classical critical point, and obtain
some useful predictions for the quantum critical point of Zφ . It is useful to
express these in terms of the dynamic susceptibility defined by
Z
Z ∞ Dh
iE
i
d
e−ikx+iωt .
(13)
χ(k, ω) =
d x
dt φ̂(x, t), φ̂(0, 0)
~
T
0
Here φ̂ is the Heisenberg field operator corresponding to the path integral
in Eqn. (12), the square brackets represent a commutator, and the angular
brackets an average over the partition function at a temperature T . The
structure of χ can be deduced from the knowledge that the quantum correlators of Zφ are related by analytic continuation in time to the corresponding
correlators of the classical statistical mechanics problem in d + 1 dimensions.
The latter are known to diverge at the critical point as ∼ 1/p2−η where p
is the (d + 1)−dimensional momentum, η is defined to be the anomalous
dimension of the order parameter (η = 1/4 for the quantum Ising chain).
Knowing this, we can deduce the form of the quantum correlator in Eq. (13)
at the zero temperature quantum critical point
χ(k, ω) ∼
1
; T = 0, g = gc .
(c2 k 2 − ω 2 )1−η/2
(14)
The most important property of Eq. (14) is the absence of a quasiparticle
pole in the spectral density. Instead, Im (χ(k, ω)) is non-zero for all ω > ck,
reflecting the presence of a continuum of critical excitations. Thus the stable
quasiparticles found at low enough energies for all g 6= gc are absent at the
quantum critical point.
We now briefly discuss the nature of the phase diagram for T > 0 with g
near gc . In general, the interplay between quantum and thermal fluctuations
near a quantum critical point can be quite complicated [h], and we cannot
discuss it in any detail here. However, the physics of the quantum Ising
chain is relatively simple, and also captures many key features found in more
complex situations, and is summarized in Fig 5. For all g 6= gc there is a
range of low temperatures (T . |g − gc |) where the long time dynamics can
be described using a dilute gas of thermally excited quasiparticles. Further,
the dynamics of these quasiparticles is quasiclassical, although we reiterate
Quantum Phase Transitions
12
T
Quantum
critical
Domain wall
quasiparticles
0
Flipped-spin
quasiparticles
gc
g
Figure 5: Nonzero temperature phase diagram of HI . The ferromagnetic order
is present only at T = 0 on the shaded line with g < gc . The dashed lines at
finite T are crossovers out of the low T quasiparticle regimes where a quasiclassical
description applies. The state sketched on the paramagnetic side used the notation
|→ij = 2−1/2 (|↑ij + |↓ij ) and |←ij = 2−1/2 (|↑ij − |↓ij ).
that the nature of the quasiparticles is entirely distinct on opposite sides of
the quantum critical point. Most interesting, however, is the novel quantum
critical region, T & |g − gc |, where neither quasiparticle picture nor a quasiclassical description are appropriate. Instead, we have to understand the
influence of temperature on the critical continuum associated with Eq. (14).
This is aided by scaling arguments which show that the only important frequency scale which characterizes the spectrum is kB T /~, and the crossovers
near this scale are universal i.e. independent of specific microscopic details
of the lattice Hamiltonian. Consequently, the zero momentum dynamic susceptibility in the quantum critical region takes the following form at small
frequencies:
1
1
.
(15)
χ(k = 0, ω) ∼ 2−η
T
(1 − iω/ΓR )
This has the structure of the response of an overdamped oscillator, and the
damping frequency, ΓR , is given by the universal expression
³
π ´ kB T
(16)
ΓR = 2 tan
16
~
The numerical proportionality constant in Eqn. (16) is specific to the quantum Ising chain; other models also obey Eqn. (16) but with a different nu-
Quantum Phase Transitions
13
merical value for this constant.
4
Beyond LGW theory
The quantum transitions discussed so far have turned to have a critical theory
identical to that found for classical thermal transitions in d + 1 dimensions.
Over the last decade it has become clear that there are numerous models,
of key physical importance, for which such a simple classical correspondence
does not exist. In these models, quantum Berry phases are crucial in establishing the nature of the phases, and of the critical boundaries between them.
In less technical terms, a signature of this subtlety is an important simplifying feature which was crucial in the analyses of Section 2: both models
had a straightforward g → ∞ limit in which we were able to write down a
simple, non-degenerate, ground state wavefunction of the ‘disordered’ paramagnet. In many other models, identification of the ‘disordered’ phase is not
as straightforward: specifying absence of a particular magnetic order as in
(2) is not enough to identify a quantum state, as we still need to write down
a suitable wavefunction. Often, subtle quantum interference effects induce
new types of order in the ‘disordered’ state, and such effects are entirely
absent in the LGW theory.
An important example of a system displaying such phenomena is the S = 1/2
square lattice antiferromagnet with additional frustrating interactions. The
quantum degrees of freedom are identical to those of the coupled dimer antiferromagnet, but the Hamiltonian preserves the full point-group symmetry
of the square lattice:
X
¡
¢
Jjk σ̂jx σ̂kx + σ̂jy σ̂ky + σ̂jz σ̂kz + . . .
(17)
Hs =
j<k
Here the Jjk > 0 are short-range exchange interactions which preserve the
square lattice symmetry, and the ellipses represent possible further multiple
spin terms. Now imagine tuning all the non-nearest-neighbor terms as a
function of some generic coupling constant g. For small g, when Hs is nearly
the square lattice antiferromagnet, the ground state has antiferromagnetic
order as in Fig 4 and Eqn. (11). What is now the ‘disordered’ ground state
for large g? One natural candidate is the spin-singlet paramagnet in Fig 2.
However, because all nearest neighbor bonds of the square lattice are now
Quantum Phase Transitions
14
or
Figure 6: Phase diagram of Hs . Two possible VBS states are shown: one which
is the analog Fig 2, and the other in which spins form singlets in a plaquette
pattern. Both VBS states have a four-fold degeneracy due to breaking of square
lattice symmetry. So the novel critical point at g = gc (described by Zz ) has the
antiferromagnetic and VBS orders vanishing as it is approached from either side:
this co-incident vanishing of orders is generically forbidden in LGW theories.
equivalent, the state in Fig 2 is degenerate with 3 other states obtained by
successive 90 degree rotations about a lattice site. In other words, the state in
Fig 2, when transferred to the square lattice, breaks the symmetry of lattice
rotations by 90 degrees. Consequently it has a new type of order, often called
valence-bond-solid (VBS) order. It is now believed [i] that a large class of
models like Hs do indeed exhibit a second-order quantum phase transition
between the antiferromagnetic state and a VBS state—see Fig 6. Both the
existence of VBS order in the paramagnet, and of a second-order quantum
transition, are features that are not predicted by LGW theory: these can only
be understood by a careful study of quantum interference effects associated
with Berry phases of spin fluctuations about the antiferromagnetic state.
We will now review the manner in which Berry phases lead to a breakdown
of LGW field theory. We begin with the field theory Zφ in Eq. (12) for
the coupled dimer antiferromagnet, and modify it to include Berry phases
of the spin. For each spin, the partition function acquires a phase factor
eiA/2 , where A is the area enclosed by the world-line of the spin on the
unit sphere in spin space. To include this contribution, it is necessary to
rewrite Eq. (12) in terms of a “hard-spin” unit vector field n, rather than the
Quantum Phase Transitions
15
soft-spin field φα . The direction of n then represents the local orientation
of the antiferromagnetic order parameter. Furthermore, the Berry phase
contributions oscillate rapidly from site to site, and so we have to write them
down on the underlying lattice, and cannot directly take the continuum limit.
In this manner, we obtain from Eq. (12)
· X Z
Z
i
2
Zn =
Dn(r, τ )δ(n (r, τ ) − 1) exp
ηj dτ Aτ (n(rj , τ ))
2 j
¸
Z
¡
¢
1
2
2
2
2
−
d rdτ (∂τ n) + c (∇r n) ,
(18)
2gc
Excluding the first Berry phase term, this is the action of the so-called O(3)
non-linear sigma model in 3 spacetime dimensions. Here we are primarily
interested in the consequences of the Berry phases: Aτ (n(τ ))dτ is defined
to be the oriented area of the spherical triangle defined by n(τ ), n(τ + dτ ),
and an arbitrary reference point n0 (which is usually chosen to be the north
pole).
The theory (18) can be considered to be the “minimal model” of quantum
antiferromagnets on the square lattice. At small g there is the conventional
magnetically ordered “Néel” phase with hni 6= 0, while at large g there is
a “quantum disordered” paramagnetic phase which preserves spin rotation
invariance with hni = 0. We are especially interested here in the nature of
this paramagnetic state.
The key to an analysis of the large g regime is a better understanding of
the nature of Aτ . We will see that Aτ behaves in many respects like the
time-component of a compact U(1) gauge field, and indeed, this accounts for
the suggestive notation. All physical results should be independent of the
choice of the reference point n0 , and it is easy to see by drawing triangles on
the surface of a sphere that changes in n0 amount to gauge transformations
of Aτ . If we change n0 to n00 , then the resulting A0τ is related to Aτ by
A0τ = Aτ − ∂τ φ(τ )
(19)
where φ(τ ) measures the oriented area of the spherical triangle defined by
n(τ ), n0 , and n00 . Furthermore, as we will discuss more completely below,
the area of any spherical triangle is uncertain modulo 4π, and this accounts
for the ‘compactness’ of the U(1) gauge theory.
Quantum Phase Transitions
16
We proceed with our analysis of Zn . First, we discretize the gradient terms
of the O(3) sigma model. We will limit our considerations here to antiferromagnets on the square lattice, but similar considerations apply to other
bipartite lattices. We also discretize the imaginary time direction, and (by
a slight abuse of notation) use the same index j to refer to the sites of a 3
dimensional cubic lattice in spacetime. On such a lattice we can rewrite (18)
as
Ã
!
Z Y
X
X
1
i
Zn =
dnj δ(n2j − 1) exp
nj · nj+µ̂ +
ηj Ajτ ,
(20)
2g
2
j
j,µ
j
where the sum over µ extends over the 3 spacetime directions, and Ajµ is
defined to equal the oriented area of the spherical triangled formed by nj ,
nj+µ and the arbitrary (but fixed) reference point n0 . We have also dropped
unimportant factors of the lattice spacing and the spin-wave velocity in (20).
The theory Eq. (20) is still cumbersome to work with because Ajτ is a complicated function of the nj . However, a purely local formulation can be found
by re-expressing nj in terms of spinor variables. We write
∗ α
njα = zja
σab zjb ,
(21)
where σ α are the Pauli matrices, the zja are two-component complex spinor
fields residing on the sites of the cubic lattice, and a is a spinor index which
extends over ↑, ↓. It is an interesting classical result in spherical trigonometry
that the area of a spherical triangle can be expressed quite simply in terms
of the spinor co-ordinates of its vertices. We will not explicitly review this
analysis here, but refer the reader to a separate review [j]. Using this result,
it is not difficult to show that Eq. (20) is very closely related to the following
partition function on the cubic lattice
Y Z 2π dAjµ Y Z
Y ¡
¢
Zz =
dzja
δ |zja |2 − 1
2π ja
jµ 0
j
Ã
!
X
¢
1 X ¡ ∗ −iAjµ
exp
z e
ηj Ajτ . (22)
zj+µ,a + c.c. + i
g jµ ja
j
Note that we have introduced a new field Ajµ , on each link of the cubic
lattice, which is integrated over. This is a compact U(1) gauge field which
Quantum Phase Transitions
17
has replaced Ajµ in Eq. (20): it can be shown [j] that the integral over Ajµ
in Eq. (22) is dominated by values Ajµ ≈ Ajµ /2, and the resulting action
differs from Eq. (20) only in unimportant details. The crucial advantage of
Eq. (22) is, of course, that there are no constraints between the zja and the
Ajµ , and we now we have to deal with a purely local lattice gauge theory.
The theory Zz now allows us to address the key questions linked to the
breakdown of LGW theory. At small g, we have, as before, a Néel state with
hza i 6= 0, and hence from Eq. (21) hni 6= 0. We will now describe the nature
of the large g paramagnetic phase, and of the transition between the small
and large g phases in the subsections below.
4.1
Nature of the paramagnet
For large g, there are strong fluctuations of the zja , and it therefore pays to
integrate out the zja from Zz and obtain an effective theory for the Ajµ . This
can be done order-by-order in 1/g in a “high temperature” expansion. The
powers of 1/g yield terms dependent upon gauge-invariant U(1) fluxes on
loops of all sizes residing on the links of the cubic lattice. For our purposes,
it is sufficient to retain only the simplest such term on elementary square
plaquettes, yielding the partition function
Ã
!
Y Z 2π dAjµ
X
1 X
ZA =
exp
cos (²µνλ ∆ν Ajλ ) + i
ηj Ajτ ,
(23)
2π
e2
jµ 0
j
¤
where ²µνλ is the totally antisymmetric tensor in three spacetime dimensions.
Here the cosine term represents the conventional Maxwell action for a compact U(1) gauge theory: it is the simplest local term consistent with the
gauge symmetry and which is periodic under Ajµ → Ajµ + 2π. The sum over
¤ in (23) extends over all plaquettes of the cubic lattice, ∆µ is the standard
discrete lattice derivative (∆µ fj ≡ fj+µ − fj for any fj ), and e2 is a coupling
constant. We expect the value of e to increase monotonically with g.
The properties of a pure compact U(1) theory have been described by Polyakov
[k]. Here we need to extend his analysis to include the all-important
Berry
R
phases in ZA . The Berry phase has the interpretation of a Jµ Aµ coupling
to a static matter field with ‘current’ Jµ = δµτ i.e. static charges ±1 on the
two sublattices. It is this matter field which will crucially control the nature
of the paramagnet.
Quantum Phase Transitions
18
1
i
1
i
-i
-1
-i
-1
1
i
1
i
-i
-1
-i
-1
Figure 7: The values of the fixed field ζi , which specify the Berry phase of the
monopole tunnelling events. The monopoles are assumed to be centered on the
sites of the dual lattice.
Polyakov showed that the quantum fluctuations of the pure compact U(1)
gauge theory were controlled by monopole tunnelling events at which the
U(1) gauge flux changed by 2π. In particular, at all values of the coupling
e, the monopoles eventually proliferate at long enough distances and lead to
confinement of ‘electric’ charges: here these electric charges are the S = 1/2
za quanta (also known as ‘spinons’).
For our purposes, we need to understand the influence of the Berry phase
terms in ZA on the monopoles. This is a subtle computation [l, m] which
has been reviewed elsewhere [j]. The final result is that each monopole can
also be associated with a Berry phase factor. If m†j is the monopole creation
at site j, then this appears in the partition function as
m†j ζj
(24)
where ζj is a fixed field taking the values 1, i, -1, −i on the four square
sublattices as shown in Fig 7.
An important consequence of these Berry phases is that the monopole operator now transforms non-trivially under the operations of the square lattice
space group. Indeed, the partition function of the antiferromagnet must be
invariant under all space group operations, and so by demanding the invariance of Eq. (24), we deduce the transformation properties of the monopole
operator. A simple analysis of Eq. (24) then shows that
Tx : m → im†
Ty : m → −im†
Quantum Phase Transitions
19
dual
Rπ/2
: m → m†
Ixdual : m → m
T : m → m.
(25)
dual
Here Tx,y are translations by one lattice spacing along the x.y axes, Rπ/2
is
dual
a rotation by π/2 about a site of the dual lattice, Ix is reflection about the
y axis of the dual lattice, and T is time-reversal.
The transformation properties in Eq. (25) now allow us to relate the monopole operator to physical observables by searching for combinations of spin
operators which have the same signature under space group operations. It
turns out that the monopole operator is connected to the VBS order parameter [m]. From Fig 6, we note that VBS order is associated with modulations
in the value of the nearest-neighbor spin-singlet correlations. So we can define
a complex order parameter, ψVBS , such that
X
α
Re [ψVBS ] = (−1)jx
σ̂jα σ̂j+x
α
jy
Im [ψVBS ] = (−1)
X
α
σ̂jα σ̂j+y
.
(26)
α
It is now easy to work out the space group transformations of ψVBS . These
lead to the important correspondence [m, i]
m ∼ e−iπ/4 ψVBS .
(27)
We have now assembled all the ingredients necessary to describe the interplay between the monopole dynamics and Berry phases in the paramagnetic
phase. Using a mapping from the compact U(1) gauge theory to a dual effective action for monopoles, the proliferation of monopoles can be argued
[j] to be equivalent to their “condensation” with hmj i 6= 0. This argument
also applies in the presence of Berry phases, although cancellations among
the phases now leads to a significantly smaller value of hmj i. Nevertheless,
it can be shown that hmj i is non-zero in the paramagnetic phase. Because
of the non-trivial transformation properties of mj under the square lattice
space group noted above, it is then clear that a non-zero hmj i spontaneously
breaks the space group symmetry. Indeed, the connection in Eq. (27) shows
that this broken symmetry is reflected in the appearance of VBS order. The
precise configuration of the VBS order depends upon the value of Arg[hmi].
Quantum Phase Transitions
20
Using Eq. (27) and the space group transformations above, the two VBS configurations shown in Fig. 6 appear for Arg[hmi] equal to π/4, 3π/4, 5π/4, 7π/4
or 0, π/2, π, 3π/2.
We have now established the breakdown of LGW theory induced by the Berry
phases in Zn in Eq. (18). A theory of the quantum fluctuations of the LGW
antiferromagnetic order parameter n does not lead to featureless ‘quantum
disordered’ paramagnetic state. Rather, subtle quantum interference effects
induce a new VBS order parameter and an associated broken symmetry.
4.2
Deconfined criticality
We now turn to a brief discussion of the quantum phase transition between
the small g Néel phase with hni 6= 0 and hψVBS i = 0 and the large g paramagnetic phase with hni = 0 and hψVBS i 6= 0. The two phases are characterized
with two apparently independent order parameters, transforming very differently under spin and lattice symmetries. Given these order parameters,
LGW theory predicts that there can be no direct second-order phase transition between them, except with fine tuning.
Recent work by Senthil et al. [i] has shown that this expectation is incorrect.
Central to their argument is the demonstration that at a possible quantum
critical point, the monopole Berry phases in Eq. (24) lead to complete cancellation of monopole effects even at the longest distance scales. Recall that
in the g > gc paramagnetic phase, Berry phases did lead to a partial cancellation of monopole contributions, but a residual effect was present at the
longest scales. In contrast, the monopole suppression is complete at the
g = gc quantum critical point.
With the suppression of monopoles, the identification of the continuum critical theory turns out to be quite straightforward. We simply treat Ajµ as
a non-compact U(1) gauge field, and take the naive continuum limit of the
action Zz in Eq. (22) while ignoring both monopoles and their Berry phases.
This leads to the field theory
µ Z
Z
h
2
Zzc = Dza (x, τ )DAµ (x, τ ) exp − d xdτ |(∂µ − iAµ )za |2 + s|za |2
i¶
1
u
2 2
2
.
(28)
+ (|za | ) + 2 (²µνλ ∂ν Aλ )
2
2e
Quantum Phase Transitions
21
In comparing Zzc to the continuum theory Zφ for the coupling dimer antiferromagnet, note that the vector order parameter φα has been replaced by a
α
spinor za , and these are related by φα ∼ za∗ σab
zb , from Eq. (21). So the order
parameter has fractionalized into the za spinons. A second novel property of
Zz is the presence of a U(1) gauge field Aµ : this gauge field emerges near the
critical point, even though the underlying model in Eqn (17) only has simple
two spin interactions.
Studies of fractionalized critical theories like Zzc in other models with spin
and/or charge excitations is an exciting avenue for further theoretical research, and promises to have significant applications to a variety of correlated
electron systems [n, o].
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